By Shafarevich I.R.

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Integrable Hamiltonian systems It may also be depictured as follows. 48 For the Heisenberg algebra the Lie-Poisson structure can be written {x,y} {x,z} {y,z} x. As above one finds that the algebra of Casimirs is given C[x], and again its spectrum can be identified with C (with coordinate u) by evaluation the Casimir x. 32 needs not hold for level sets). It follows that p = RS 3 + S 2 and = =0, = as by on an all Pc = { RS 2 if u(c) =/ 0, S 2 if u(c) = 0. This case is depictured as follows. 49 An interesting example is found by taking the Lie-Poisson structure on g* = gl(2)*.

Therefore we will call Spec Cas M the parameter space and the map M-.. SpecCasM the parameter map of the affine Poisson variety (M,{-,·}). The fibers of the parameter map will also be called level sets of the Casimirs because picking a fiber (over a closed point) corresponds to fixing some (all) Casimirs. The decomposition of M into the fibers over closed points is called the Casimir decomposition. 14 below). 2) Since Cas(M) is a Poisson subalgebra of O(M) (actually a Lie ideal), Poisson morphism, Spec Cas( M) having the trivial Poisson structure.

4 ). In terms of coordinates { t 1 , ... 1) that Cas(C 6 ) = C[t 1 t 2 t 3 ,t 4 + t 5 + t 6], so that SpecCas(C 6 ) can be identified with C 2 (with coordinates u and v) by evaluating a homomorphism on t1 tzt3 resp. on t4 +ts +t6 (in that order). By computing a few determinants one sees that the rank is zero on the three-plane t 1 = t 2 = t 3 = 0, two on the three four-planes t; = t j = 0 (1 ~ i < j ~ 3) and four elsewhere. From it one easily obtains the following invariant polynomials: p = R 2 S6 Pc = { + 3RS 4 +5 3 , R2S4 for u(c) 3R 2 S 4 + f= 3RS 3 0, + S2 for u(c) = 0.

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